A384033 a(n) is the number of solutions to n = sopfr(k*sopfr(n)) where sopfr(m) is sum of prime factors of m counted with multiplicity.

0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 2, 1, 3, 4, 5, 1, 7, 1, 10, 10, 4, 1, 19, 19, 19, 30, 17, 1, 40, 1, 52, 46, 12, 77, 87, 1, 77, 111, 87, 1, 175, 1, 197, 157, 197, 1, 219, 302, 413, 372, 175, 1, 413, 614, 413, 456, 67, 1, 1083, 1, 677, 819, 1552, 1552, 1695, 1

Offset: 1

Author

Michael Terhoeven, May 17 2025

Keywords

• nonn

Comments

Also, a(n) is the number of star graph labelings with label n on the leaves of the star such that the label of every vertex equals the sum of prime factors (with repetition) over adjacent vertices and all labels are positive.

Examples

			A star graph can be defined differently; here, the star graph S_k has a total of k vertices and k-1 leaves.
For all primes n=p as leaf label, the star graph S_2 with 1 central vertex labeled p and 1 leaf labeled p is a solution and the only solution; thus, a(p)=1.
  p - p
n=4 as leaf label, works as well with S_2 (2+2=2*2); thus, a(4)=1.
  4 - 4
For n=6 as leaf label, there is no respective star graph (as the sum of prime factors is 2+3=5).
n=8 as leaf label can be done as S_4 with the center labeled 18 and the leaves labeled 8 (the prime factor sum for 18=2*3^2 is 2+3+3=8; the prime factor sum for 8=2^3 is 6, adding over all 3 leaves, results in 18); thus, a(8)=1.
       8
       |
  8 -  18 - 8
n=12 is the first labeling with 2 solutions: it can be done with central vertex labeled 35 (S_6) and 42 (S_7); thus, a(12)=2.
  The S_6 solution looks like
       12   12
        \  /
    12 - 35 - 12
          |
         12
  as 12=2*2*3 -> 7, 5 leaves sum up to 35 = 5*7 -> 12.
		

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..10000
• Sean A. Irvine, Java program (github)
• Eric Weisstein's World of Mathematics, Star graph
• Wikipedia, Star (graph theory)
• Robert G. Wilson v, n, a(n) & center of the star graph

Crossrefs

Cf. A000607, A001414.

Programs

Mathematica

sopfr[n_] := sopfr[n] = Plus @@ Times @@@ FactorInteger@ n; sopfr[1] = 0; t = Series[1/Product[1 - x^Prime[i], {i, 100}], {x, 0, 100}]; a[n_] := Block[{s = sopfr@sopfr@n}, If[s <= n, Coefficient[t, x, n - s], 0]]; Array[a, 67] (* - Robert G. Wilson v, Jul 16 2025 *)

Formula

a(n) = A000607(n - sopfr(sopfr(n))) if sopfr(sopfr(n)) <= n, otherwise a(n)=0. - Sean A. Irvine, Jun 27 2025

a(p) = 1 if p is a prime. Also, a(n) = 1 when n = {4, 8, 9, 10}. - Robert G. Wilson v, Jul 16 2025

Extensions

More terms from Sean A. Irvine, Jun 27 2025